Determining r- and (r,s)-Robustness of Digraphs Using Mixed Integer Linear Programming \thanksreffootnoteinfo

Determining - and -Robustness of Digraphs Using Mixed Integer Linear Programming \thanksreffootnoteinfo

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Abstract

There has been an increase in the use of resilient control algorithms based on the graph theoretic properties of - and -robustness. These algorithms guarantee consensus of normally behaving agents in the presence of a bounded number of arbitrarily misbehaving agents if the values of the integers and are sufficiently large. However, determining an arbitrary graph’s robustness is a highly nontrivial problem. This paper introduces a novel method for determining the - and -robustness of digraphs using mixed integer linear programming; to the best of the authors’ knowledge it is the first time that mixed integer programming methods have been applied to the robustness determination problem. The approach only requires knowledge of the graph Laplacian matrix, and can be formulated with binary integer variables. Mixed integer programming algorithms such as branch-and-bound are used to iteratively tighten the lower and upper bounds on and . Simulations are presented which compare the performance of this approach to prior robustness determination algorithms.

Umich]James Usevitch, Umich]Dimitra Panagou,

Department of Aerospace Engineering, University of Michigan 


Key words:  Networks, Robustness, Graph Theory, Optimization problems, Integer programming

 

11footnotetext: The authors would like to acknowledge the support of the Automotive Research Center (ARC) in accordance with Cooperative Agreement W56HZV-14-2-0001 U.S. Army TARDEC in Warren, MI, and the Award No W911NF-17-1-0526.

1 Introduction

Consensus on shared information is fundamental to the operation of multi-agent systems. In context of mobile agents, it enables formation control, rendezvous, distributed estimation, and many more objectives. Although a vast literature of algorithms for consensus exist, many are unable to tolerate the presence of misbehaving agents subject to attacks or faults. Recent years have seen an increase of attention on resilient algorithms that are able to operate despite such misbehavior. Many of these algorithms have been inspired by [9], which is one of the seminal papers on consensus in the presence of adversaries; [15, 41, 14], which outline discrete- and continuous-time algorithms along with necessary and sufficient conditions for scalar consensus in the presence of Byzantine adversaries; and [37, 36, 31, 33], which outline algorithms for multi-agent vector consensus of asynchronous systems in the presence of Byzantine adversaries. Some of the most recent resilience-based results that draw upon these papers include state estimation [18, 20, 19, 21], rendezvous of mobile agents [23, 22], output synchronization [11], simultaneous arrival of interceptors [17], distributed optimization [30, 28], reliable broadcast [32, 41], clock synchronization [7, 8], randomized quantized consensus [5], self-triggered coordination [25], and multi-hop communication [29].

A large number of results on network resilience are based upon the graph theoretical properties known as -robustness and -robustness [15, 41]. -robustness and -robustness are key notions included in the sufficient conditions for convergence of several resilient consensus algorithms including the ARC-P [14], W-MSR [15], SW-MSR [24], and DP-MSR [4] algorithms. Given an upper bound on the global or local number of adversaries in the network, the aforementioned resilient algorithms guarantee convergence of normally behaving agents’ states to a value within the convex hull of initial states if the integers and are sufficiently large.

A key challenge in implementing these resilient algorithms is that determining the - and -robustness of arbitrary digraphs is an NP-hard problem in general [13, 40]. The first algorithmic analysis of determining the values of and for arbitrary digraphs was given in [13]. The algorithms proposed in [13] employ an exhaustive search to determine the maximum values of and for a given digraph, and have exponential complexity w.r.t. the number of nodes in the network. In [40] it was shown that the decision problem of determining if a graph is -robust for a given integer is coNP-complete. Subsequent work has focused on methods to circumvent this difficulty, including graph construction methods which increase the graph size while preserving given values of and [15, 6]; demonstrating the behavior of as a function of particular graph properties [40, 27, 42]; and lower bounding with the isoperimetric constant and algebraic connectivity of undirected graphs [26]. The majority of these approaches either consider only undirected graphs, or specific classes of directed graphs with particular properties. Another approach has recently used machine learning to correlate characteristics of certain graphs to the values of and [38], but these correlations are inherently stochastic in nature and do not provide explicit guarantees. Finding more efficient or practical ways of determining the robustness of graphs in general, and digraphs in particular, is still an open problem.

In this paper, we introduce novel methods for determining the - and -robustness of digraphs using mixed integer linear programming (MILP). These methods only require knowledge of the graph Laplacian matrix and are zero-one MILPs, i.e. with all integer variables being binary. To the best of our knowledge, this is the first time the robustness determination problem has been formulated in this way. These results connect the problem of graph robustness determination to the extensive and well-established literature on integer programming and linear programming.

1.1 Contributions

This paper makes the following contributions:

  1. We present a method to determine the maximum integer for which a nonempty, nontrivial, simple digraph is -robust using mixed integer linear programming.

  2. We present a method which determines the -robustness of a digraph using linear programming. Here, the -robustness of a digraph refers to the maximal integer pair according to a lexicographical order for which a given digraph is -robust, as first described in [13]. Furthermore, we show that our method can also determine the maximum integer for which a digraph is -robust, which is not considered in [13].

  3. We present two mixed integer linear programs whose optimal values provide lower and upper bounds on the maximum for which a nonempty, nontrivial, simple digraph is -robust. These two formulations exhibit a lower complexity than the method in the first contribution described above.

The contributions of this paper provide several advantages. First, expressing the robustness determination problem in MILP form allows for approximate lower bounds on a given digraph’s -robustness to be iteratively tightened using algorithms such as branch-and-bound. Lower bounds on the maximum value of for which a given digraph is robust (for a given nonnegative integer ) can also be iteratively tightened using the approach in this paper. Prior algorithms are only able to tighten the upper bound on the maximum robustness for a given digraph or undirected graph. Second, this formulation enables commercially available solvers such as Gurobi or MATLAB’s intlinprog to be used to find the maximum robustness of any digraph. Finally, experimental results using this new formulation suggest a reduction in computation time as compared to the centralized algorithm proposed in [13].

Part of this paper was previously submitted as a conference paper [34]. The extensions to the conference version include the following:

  • The more general case of determining -robustness is considered.

  • Two optimization problems with reduced-dimension binary vector variables are given whose optimal values provide a lower bound and an upper bound, respectively, on the maximum value of for which a graph is -robust.

This paper is organized as follows: notation is presented in Section 2, the problem formulation is given in Section 3, determining the -robustness of digraphs is treated in Section 4, determining the values of for which a digraph is -robust for a given is treated in Section 5, methods to obtain upper and lower bounds on the maximum for which a digraph is -robust are presented in Section 6, a brief discussion about the advantages of the MILP formulations is given in Section 7, simulations demonstrating our algorithms are presented in Section 8, and a conclusion and discussion about future work are given in 9.

2 Notation

The sets of real numbers and integers are denoted and , respectively. The sets of nonnegative real numbers and integers are denoted and , respectively. denotes an -dimensional vector space over the field , represents the set of dimensional vectors with integer entries, and represents a binary vector of dimension . Scalars are denoted in normal text (e.g. ) while vectors are denoted in bold (e.g. ). The notation denotes the th entry of vector .

The inequality symbol denotes a componentwise inequality between vectors; i.e. for , implies . A vector of ones is denoted , and a vector of zeros is denoted , where the length of each vector will be implied by the context. The union, intersection, and set complement operations are denoted by , and , respectively. The cardinality of a set is denoted as , and the empty set is denoted . The infinity norm of a vector is denoted . The notations are both used in this paper to denote the binomial coefficient with . Given a set , the power set of is denoted . Given a function , the image of a set under is denoted . Similarly, the preimage of under is denoted . The logical OR operator, AND operator, and NOT operator are denoted by , respectively. The lexicographic cone is defined as for some . The lexicographic ordering on is defined as if and only if [3, Ch. 2].

A directed graph (digraph) is denoted as , where is the set of indexed vertices and is the edge set. This paper will use the terms vertices, agents, and nodes interchangeably. A directed edge is denoted , with , meaning that agent can receive information from agent . The set of in-neighbors for an agent is denoted . The minimum in-degree of a digraph is denoted . Occasionally, will be used to denote an undirected graph, i.e. a digraph in which . The graph Laplacian for a digraph (or undirected graph) is defined as follows, with denoting the entry in the th row and th column:

(1)

3 Problem Formulation

The notions of - and -robustness are graph theoretical properties used to describe the communication topologies of multi-agent networks. Examples of such networks include stations in a power grid, satellites in formation, or a group of mobile robots. In these networks, edges model the ability for one agent to transmit information to another agent . Prior literature commonly considers simple digraphs, which have no repeated edges or self edges [16, 15, 12, 37, 35]. More specifically, simple digraphs satisfy , and if the directed edge , then it is the only directed edge from to . Prior work also commonly considers nonempty and nontrivial graphs, where .

Assumption 1

This paper considers nonempty, nontrivial, simple digraphs.

The property of -robustness is based upon the notion of -reachability. The definitions of -reachability and -robustness are as follows:

Definition 1 ([15])

Let and be a digraph. A nonempty subset is -reachable if such that .

Definition 2 ([15])

Let . A nonempty, nontrivial digraph on nodes is -robust if for every pair of nonempty, disjoint subsets of , at least one of the subsets is -reachable. By convention, the empty graph is 0-robust and the trivial graph is 1-robust.

The property of -robustness is based upon the notion of -reachability. The definitions of -reachability and -robustness are as follows:

Definition 3 ([15])

Let be a nonempty, nontrivial, simple digraph on nodes. Let , . Let be a nonempty subset of , and define the set . We say that is an -reachable set if there exist nodes in , each of which has at least in-neighbors outside of . More explicitly, is -reachable if .

Definition 4 ([15])

Let , . Let be a nonempty, nontrivial, simple digraph on nodes. Define , . The digraph is -robust if for every pair of nonempty, disjoint subsets , at least one of the following conditions holds:

  1. ,

  2. ,

  3. .

The properties of - and -robustness are used to quantify the ability of several resilient consensus algorithms to guarantee convergence of normally behaving agents in the presence of Byzantine and malicious adversaries, collectively referred to in this paper as misbehaving agents [15, 11, 4, 24, 14]. Larger values of and generally imply the ability of networks applying these resilient algorithms to tolerate a greater number of misbehaving agents in the network. For a more detailed explanation of the properties of -robustness and -robustness, the reader is referred to [16, 15, 40].

When considering a particular digraph , there may be multiple values of for which is -robust. Similarly, there may be multiple values of and for which is -robust. The following properties of robust graphs demonstrate this characteristic:

Property 1 ([10], Prop. 5.13)

Let be an arbitrary, simple digraph on nodes. Suppose is -robust with and . Then is also -robust and .

Property 2 ([10], Prop. 5.20)

Let be an arbitrary, simple digraph on nodes. Suppose is -robust with and . Then is -robust.

As an example, if a digraph is -robust, then by Property 1 it is also simultaneously 3-robust, 2-robust, and 1-robust. In addition, if a digraph is -robust, then it is simultaneously -robust for all integers and . Moreover, by Property 2, is also -robust, -robust, -robust, and -robust. For notational purposes, we denote the set of all values for which a digraph is -robust as , where . By Definition 4, is explicitly defined as

(2)

Note that the conditions of (3) are simply an alternate way of expressing the conditions of Definition 4.

To characterize the resilience of graphs however, prior literature has generally been concerned with only a few particular values of and for which a given digraph is - or -robust. For -robustness, the value of interest is the maximum integer for which the given digraph is -robust.

Definition 5

We denote the maximum integer for which a given digraph is -robust as .

Several resilient algorithms guarantee convergence of the normal agents when the adversary model is -total or -local in scope,\@footnotemark\@footnotetextAn -total adversary model implies that there are at most misbehaving agents in the entire network. An -local adversary model implies that each normal agent has at most misbehaving agents in its in-neighbor set. and the digraph is -robust. The value of therefore determines the maximum adversary model under which these algorithms can operate successfully. Furthermore, all other values of for which a digraph is -robust can be determined from by using Proposition 1.

For -robustness, there are two pairs of interest. The authors of [13] order the elements of using a lexicographical total order, where elements are ranked by value first and value second. More specifically, if and only if , where is the lexicographic cone defined in Section 2. Their algorithm finds the maximum element of with respect to this order. For notational clarity, we denote this maximum element as .

Definition 6

Let be defined as in (3). The element is defined as the maximum element of under the lexicographical order on .

The other pair of interest is , where . Several resilient algorithms guarantee convergence of the normally behaving agents when the (malicious [15]) adversary model is -total in scope and the digraph is -robust. The value determines the maximum malicious adversary model under which these algorithms can operate successfully. The value of does not always coincide with the -robustness of the digraph. A simple counterexample is given in Figure 1, where the -robustness of the graph is but the value of is equal to .

The purpose of this paper is to present methods using mixed integer linear programming to determine , the -robustness of , and the -robustness of for any nonempty, nontrivial, simple digraph .

Problem 1

Given an arbitrary nonempty, nontrivial, simple digraph , determine the value of .

Problem 2

Given an arbitrary nonempty, nontrivial, simple digraph , determine the -robustness of .

Problem 3

Given an arbitrary nonempty, nontrivial, simple digraph , determine the -robustness of .

3.1 Additional Notes

The values of for which a digraph can be -robust lie within the interval [10, Property 5.19]. In addition, -robustness is equivalent to -robustness [13, Property 5.21], [15, Section VII-B] which implies that the values of for which a graph can be -robust fall within the same interval. The values of for which a digraph can be -robust lie within the interval .\@footnotemark\@footnotetextFootnote 8 in [15] offers an excellent explanation for restricting to this range by convention. However, we will use an abuse of notation by denoting a graph as -robust for a given if the graph is not -robust.

Fig. 1: An example of the elements of for a digraph . Since , the possible values of and for which the digraph is -robust fall within the range , . One possible pair of subsets and is depicted, which satisfies , , and . By Definition 4, therefore cannot be -robust, -robust, or -robust.

It should be clear from Definitions 2 and 4 that determining and -robustness for a digraph by using an exhaustive search method is a combinatorial problem, which involves checking the reachabilities of all nonempty, disjoint subsets of . For notational purposes, we will denote the set of all possible pairs of nonempty, disjoint subsets of as . More explicitly, is defined as

(3)

It was shown in [13] that .\@footnotemark\@footnotetextSince , the total number of unique nonempty, disjoint subsets is , denoted as in [13]. As a simple example, Figure 2 depicts all elements of for a graph of 3 agents, i.e. all possible ways to choose two nonempty, disjoint subsets from the graph.

Fig. 2: Depiction of all 12 possible elements in for a complete graph of 3 agents. Each graph represents a different possible way of dividing into sets and . In each individual graph, yellow agents are in , blue agents are in , and white agents are in neither nor .

4 Determining -Robustness using Mixed Integer Linear Programming

In this section we will demonstrate a method for solving Problem 1 using a mixed integer linear program (MILP) formulation. An MILP will be presented whose optimal value is equal to for any given nonempty, nontrivial, simple digraph .

First, an equivalent way of expressing the maximum robustness of a digraph is derived. This equivalent expression will clarify how can be determined by means of an optimization problem. Given an arbitrary, simple digraph and a subset , the reachability function is defined as follows:

(4)

In other words, the function returns the maximum for which is -reachable. Using this function, the following Lemma presents an optimization formulation which yields :

Lemma 1

Let be an arbitrary nonempty, nontrivial, simple digraph with . Let be defined as in Definition 5. The following holds:

subject to (5)

PROOF. Note that implies or . Let be a minimizer of (1). Then . Therefore either or . This satisfies the definition of -robustness as per Definition 2, therefore is at least -robust. This implies .

We next show that . We prove by contradiction. Recall from Definition 5 that is the maximum integer for which is -robust, which means is -robust by definition. Suppose . This implies and . Since the nonempty, disjoint subsets satisfy and , by the negation of Definition 2 this implies that is not -robust. However, this contradicts the definition of being the largest integer for which is -robust (Definition 5). This provides the desired contradiction; therefore .

Remark 1

Using the definition of in (3), the constraints on the RHS of (1) can be made implicit [3, section 4.1.3] as follows:

(6)

4.1 Reformulating the Objective Function

We demonstrate next that the objective function of (1) can be expressed as a function of the network Laplacian matrix. Recall that and that represents a binary vector of dimension . The indicator vector is defined as follows: for all ,

(7)

In other words the th entry of is 1 if the node with index is a member of the set , and zero otherwise. It is straightforward to verify that is a bijection. Therefore given , the set is defined by . Finally, observe that for any , . The following Lemma demonstrates that for any , the function can be determined as an affine function of the network Laplacian matrix and the indicator vector of :

Lemma 2

Let be an arbitrary nonempty, nontrivial, simple digraph, let be the Laplacian matrix of , and let . Then the following holds for all :

(8)

where is the th row of . Furthermore,

(9)

PROOF. The term is shortened to for brevity. Recall that the entry in the th row and th column of is denoted . The definition of from (2) implies

(10)

Since by (7), implies , the term . In addition, since implies , the term . By this, equation (10) simplifies to .

The value of the term depends on whether or . If , then , implying . If , then implying . This proves the result for equation (8).

To prove (9), we first consider nonempty sets . By the results above and (4), the maximum reachability of any is found by

(11)

By its definition, . Observe that if then , implying . Conversely, if an agent is not in the set , then the function takes the nonpositive value . This implies . By these arguments, we therefore have , which implies

(12)

Therefore by equations (12) and (11), the maximum reachability of is found by the expression

(13)

Lastly, if , then by (4) we have . In addition, , implying that .

Using Lemma 2, it will next be shown that the objective function of of (1) can be rewritten as the maximum over a set of affine functions:

Lemma 3

Consider an arbitrary, nonempty, nontrivial, simple digraph . Let be the Laplacian matrix of , and let be the th row of . Let be defined as in (3). Then for all the following holds:

(14)

PROOF. By Lemma 2, for , and for . The result follows.

From Lemma 1, Lemma 3, and Remark 1, we can immediately conclude that satisfies

(15)

Note that the terms and are each -dimensional binary vectors. Letting and , the objective function of (4.1) can be written as . Every pair can be mapped into a pair of binary vectors by the function , where . By determining the image of under , the optimal value of (4.1) can be found by minimizing over pairs of binary vectors directly. Using binary vector variables instead of set variables will allow (4.1) to be written directly in an MILP form. Towards this end, the following Lemma defines the set :

Lemma 4

Let be an arbitrary nonempty, nontrivial, simple digraph, and let be defined as in (3). Define the function as

(16)

Define the set as

(17)

Then both of the following statements hold:

  1. The image of under is equal to , i.e.

  2. The mapping is a bijection.

PROOF. We prove 1) by showing first that , and then . Any satisfies , , as per (3). Observe that

Because and , then . Otherwise if then either or , which both contradict the definition of . Therefore , and by similar arguments . Observe that

Finally, implies that and . Therefore and . This implies that

Therefore for all , satisfies the constraints of the set on the RHS of (17). This implies that .

Next, we show by showing that for all , there exists an such that . Choose any and define sets as follows:

(18)

For the considered sets , implies and implies . In addition since , we have and . By our choice of and , we have , and from previous arguments . Similar reasoning can be used to show that . These arguments imply that . Consequently, satisfies all the constraints of and is therefore an element of . Clearly, by (18) we have , which shows that there exists an such that . Since this holds for all , this implies . Therefore .

We now prove 2). Since , the function is surjective. To show that it is injective, consider any and such that . This implies . Note that if and only if and . Since the indicator function is itself injective, this implies and , which implies . Therefore is injective.

Using Lemma 4 allows us to present the following mixed integer program which solves Problem 1:

Theorem 1

Let be an arbitrary nonempty, nontrivial, simple digraph and let be the Laplacian matrix of . The maximum -robustness of , denoted